Time resolution of digital sampling

[pkane2001]

Just now, manueljenkin said:

Only for any "truly periodic" signal. If he has a sawtooth running from -infinity to +infinity, yeah. Otherwise, no!

 

A finite-length signal is easy to handle as if it's infinite by windowing in the time domain. And before you bring up tradeoffs of window functions as an issue, there are some very good windows available that have tiny effects on time domain at the limits of floating point calculations and well below any audibility levels (-200dB or less). So yes, nothing is perfect in the real world, but it can be as perfect as you'd like it to be, as long as you still keep it finite.

[manueljenkin]

13 minutes ago, pkane2001 said:

 

A finite-length signal is easy to handle as if it's infinite by windowing in the time domain. And before you bring up tradeoffs of window functions as an issue, there are some very good windows available that have tiny effects on time domain at the limits of floating point calculations and well below any audibility levels (-200dB or less). So yes, nothing is perfect in the real world, but it can be as perfect as you'd like it to be, as long as you still keep it finite.

Not if I have transients existing in my signal. It can modulate anything to any large extent! This 200db dip may exist that way for a normal infinite period sinusoid whose components won't leak out much, but not so for transients. The transients could modulate the sampled signals heavily even on a perfect low pass, let alone an imperfect one.

 

Also this -200db scaling would mean nothing if the amplitude of noise/signal/component of transient at that frequency is very high. The end result would be amplitude of the signal * scaling factor. If it is large enough it'll be able to comfortably modulate the signal by a significant amount.

[pkane2001]

5 minutes ago, manueljenkin said:

Not if I have transients existing in my signal. It can modulate anything to any large extent! This 200db dip may exist that way for a normal infinite period sinusoid whose components won't leak out much, but not so for transients. The transients could modulate the sampled signals heavily even on a perfect low pass, let alone an imperfect one.

Are you still talking about infinite slope transients?

[jabbr]

5 minutes ago, pkane2001 said:

Are you still talking about infinite slope transients?

:SMH:

🤷‍♂️

[manueljenkin]

22 minutes ago, pkane2001 said:

Are you still talking about infinite slope transients?

If you want to fix a bound, make a study, show me the bound of naturally existing signal transients, we can then work it out. Else I think I can keep the unit step function as a guard limit. As long as there is no "conclusive" study it shall remain this way.

 

I'm not really fixing it into infinite band/slew transients as much as I'm into understanding the real limits of transients and the possibilities of them being able to modulate the output at any given sampling limit using Nyquist Shannon sampling procedure. Especially the existing 44.1khz sampling rate. You can't just push what you "think" as a conclusive fact, and ask us to belive it. If you or anyone has truly validated the real world scenarios, kindly show us the analysis/numbers and coverage metrics.

[manueljenkin]

4 hours ago, manueljenkin said:

For those who haven't been able to follow till now. You can look into the derivations below. @Jud This one is for your request mainly.

 

I have taken an "ideal" infinite time sinc filter which has the best frequency response of a rectangular function. So this is an ideal scenario, in reality we only have truncated or windowed sinc function and I have already drawn and shown what happens if I convolve one over another for these non ideal functions (the modulations/deviations grow much larger every iteration)

 

I am just multiplying them in frequency domain to see how the results come out.

 

Formula references here: https://math.stackexchange.com/questions/736749/fourier-transform-of-sinc-function

 

http://www.thefouriertransform.com/pairs/sinusoids.php

 

I have done the operation of ideal low passing on steady state sines that begin at time -infinity and end at time +infinity (otherwise the derived Fourier transform won't be the same as above, and you'll have deviations/alias components at the filtered output) and squares (again repetitive pulses from -infinity to +infinity) till input 4 and I have shown how you can perfectly bandlimit it if you can have such an ideal scenario with ideal infinite time low pass. Whatever phase deviations I have from the ideal zero point will still be preserved in the Fourier transform and I can extract it very well since passing them through an ideal low pass either passes them directly, or completely truncates their component. Provided I have high enough bit depth I can precisely say the delta time. When I don't have enough bit depth the formula described in the first post of this thread comes into play. All of these are assuming steady state signals having components of time -infinity to infinity. So far so good.

Real truncated low pass as I said will leak some content out of the spectral band, but let's keep it aside for now.

 

Now the fun comes in when you use a real transient scenario, change from one periodic/a-periodic state to another. One of the best examples to show this transient scenario from inertia is the heaviside function. You can just take the heaviside function, or take a transient scenario of any periodic function which would just be the function from -infinity to + infinity multiplied piecewise over the heaviside function. Just try passing this heaviside function over the sinc low pass and let me know your results.

 

http://www.theochem.ru.nl/~pwormer/Knowino/knowino.org/wiki/Heaviside_step_function.html

 

You'll always have it leaking things out of the pass band. It doesn't even matter if these are themselves audible or not (prior to sampling), if they exist in the signal when sampling (ie input signal is not completely band limited), they will alias and influence the audible bands when sampled and reconstructed!! Some of these transients may be audible some may not be, regardless, their mere existence and inability to be completely band limited will influence additional modulations, on top of discarding potentially audible info. If you want this to be free of issues, you need to visualize the true bounds of real world transients and then optimize for it, not randomly chop at 22khz. People love to throw around the terms practically it doesn't go to "infinite slew" yeah it doesn't, but analyse and judge the true bounds in which it really occurs and also optimize the mic to the limits of human hearing (and not cause any transient noise 😉). Till then I can ask for higher sample rate to be on the safer side of data preservation. The moment you sample a non band limited waveform and introduce aliasing components, you've introduced modulations of extremely unpredictable order.

 

Also stop saying wrong conclusions and then cloaking it inside "math". Math doesn't say it. I've often see people invoke the word "science" for things it never said. They just try to push their personal opinion by falsely stating it as science. We've had our fair share of such instances in audio - especially analog circuits. Only a truly periodic function has a unique frequency. Anything that's not periodic/transient doesn't have a unique frequency. It'll change wildly depending upon how you window and visualize it, and this windowing will have trade offs in time domain visualization. I've already described this in previous posts. They don't have a "frequency", since they don't repeat.

 

If you don't like heaviside function, do some other function with real transient. You'll end up with similar dilemma.

 

Also regarding transients of synths or codes. I'm not going to comment on Synth, but a coder can intend it to have "any" transient rate. If you don't sample it properly, you end up with modulations, that's all.

Also, kindly note scenario 6 in this diagram.  Piecewise multiplication of time "x units" delayed heaviside with time "x units" delayed sine (so that it delays and begins at zero without any abrupt jump). It is also a transient scenario. It doesn't have "infinite slew" because there is no abrupt jump. Yet when you do ft and multiply with ft of even an ideal sinc filter you will still get out of band results, pretty high with reasonable windows at time "x", and again as you move and mainpulate the window the leakage changes. The leakage scenario holds here too. This is an "optimistic" scenario of transient and the scenario 5 is an "theoretical ideal" scenario of transient, reality lies somewhere in the middle. You can already see neither of them have a unique frequency, the way you window and look at them decides how much frequency and dc offset you visualize.

[PeterSt]

3 hours ago, pkane2001 said:

Are you still talking about infinite slope transients?

 

Please notice that we should not make the mistake in thinking that this will be put through to the output. So even if any stupid proposes a 1 sample Dirac pulse to the input, *any* filtering means already diminishes that to a more gradually peak. Just like I showed it.

So it is not necessary to be funny. Haha. Oops.

 

One could also attest that any Dirac (made by stupids) can only exist in digital and in the bandlimited system (domain); would the bandwidth be infinite, then the pulse very theoretically would be allowed to be infinite, but it would be useless because no hardware would be able to deal with it (just because in physics infinity does not exist).

 

In a next post I will have an attempt to make this clear better. Or to let you (guys) possibly think differently. Throw out your textbook math and such. Create new math if it is deemed necessary (that would be comfortable !).

[PeterSt]

So here I am. Me and my keyboard. I am going to synthesize some sounds. I start out with a (stupid) pulse train, each pulse one sample long. I envision pulses 1200 times per second. Because I am going to produce in RebBook CD, I choose the 16/44.1 PCM format. People of course look forward to my next album of test signals (just like my wife hates them).

Wait, I already had such a file prepared for other reasons. Good.

 

And so I am going to play back those pulses we have seen a couple of times by now. The output (once again) shows like this, measured from the output of my DAC. Btw, of the contenders in this thread, @jabbr could do the same, and let's say that @Jud already listened to it, a few years ago. 

 

 

So them again, after those one sample pulses have been processed with the 16x upsampling (filtering).

 

What do people reckon I hear from this ? A few hints:

I don't listen through headphones;

I am pretty sure that my hearing won't go beyond 12KHz these days.

 

People should try to guess right, because at least this is the most easy thing to do. Load the file and play ...

(I will *not* be able to show you what I hear from it, because it would require a destructive digital recording - you could come over though - so for now you'd have to trust me, but I know you will)

 

?

 

[manueljenkin]

1 hour ago, manueljenkin said:

Also, kindly note scenario 6 in this diagram.  Piecewise multiplication of time "x units" delayed heaviside with time "x units" delayed sine (so that it delays and begins at zero without any abrupt jump). It is also a transient scenario. It doesn't have "infinite slew" because there is no abrupt jump. Yet when you do ft and multiply with ft of even an ideal sinc filter you will still get out of band results, pretty high with reasonable windows at time "x", and again as you move and mainpulate the window the leakage changes. The leakage scenario holds here too. This is an "optimistic" scenario of transient and the scenario 5 is an "theoretical ideal" scenario of transient, reality lies somewhere in the middle. You can already see neither of them have a unique frequency, the way you window and look at them decides how much frequency and dc offset you visualize.

@PeterSt please have a look. You don't need infinite slew to denote transients. These non infinite slew rate signals would also leak signal out of band even when low passed by an ideal sinc filter, because they are still "transients". Nyquist Shannon reconstruction will struggle with these signals too since they spur out of band signals when low passed which cause aliasing.

 

The leakage is just going to get worser and worser as your slew rate at this transition point increases, but even with a fairly slow slew rate transient, there is out of band leakage when sampling using a sinc low pass filter

[PeterSt]

2 minutes ago, manueljenkin said:

@PeterSt please have a look.

 

Yes, I know. Earlier in the thread I actually showed that. In-band the same sh*t is there already, with or without sinc filter.

It would be a bit tedious to show folding back to the audio band of images beyond the Nyquist limit, although with some careful spacing I suppose it can be made visible. But careful, because the out of band images should not be ! ... otoh, I know how much down they are to begin with ... (no ready plot of than at hand, but trust me: harmless).

[pkane2001]

4 hours ago, PeterSt said:

 

Please notice that we should not make the mistake in thinking that this will be put through to the output. So even if any stupid proposes a 1 sample Dirac pulse to the input, *any* filtering means already diminishes that to a more gradually peak. Just like I showed it.

So it is not necessary to be funny. Haha. Oops.

 

One could also attest that any Dirac (made by stupids) can only exist in digital and in the bandlimited system (domain); would the bandwidth be infinite, then the pulse very theoretically would be allowed to be infinite, but it would be useless because no hardware would be able to deal with it (just because in physics infinity does not exist).

 

In a next post I will have an attempt to make this clear better. Or to let you (guys) possibly think differently. Throw out your textbook math and such. Create new math if it is deemed necessary (that would be comfortable !).

 

New math is not necessary, since the old math is perfectly adequate. Anything new you come up with must not contradict a proven mathematical theorem. It doesn't matter how many other ways you try to approach it, if your result goes against a proven theorem, your way is wrong and not the other way around.

 

[manueljenkin]

7 minutes ago, pkane2001 said:

 

New math is not necessary, since the old math is perfectly adequate. Anything new you come up with must not contradict a proven mathematical theorem. It doesn't matter how many other ways you try to approach it, if your result goes against a proven theorem, your way is wrong and not the other way around.

 

Please "prove" nyquist Shannon's sampling to work without artefacts for the transients I mentioned, using any kind of sinc filter (even infinite time). Show me how you'll perfectly band limit these transients using a sinc filter. I give you option for two scenarios - 5 and 6.

 

I've shown you the proof it won't work. Look at the ft of the functions. If you actually manage to prove it, you would have invented "new" math proof.

[pkane2001]

1 minute ago, manueljenkin said:

Please "prove" Nyquist Shannon's sampling to work without artefacts for the transients I mentioned. I give you option for two scenarios - 5 and 6.

 

I've shown you the proof it won't work. Look at the ft of the functions. If you actually manage to prove it, you would have invented "new" math proof.

 

YOU WILL NEVER REPRODUCE A PERFECT HEAVISIDE FUNCTION OR A DIRAC PULSE. PERIOD.

 

No matter how much new math you invent, this is not possible. The best thing you can do is approximate it within some bandlimit. That creates "artifacts", such as Gibbs, that are much, much larger than anything a proper windowed FFT would cause. Accept that real world cannot be perfect, and an ideal pulse is not possible. 

[manueljenkin]

4 minutes ago, pkane2001 said:

 

YOU WILL NEVER REPRODUCE A PERFECT HEAVISIDE FUNCTION OR A DIRAC PULSE. PERIOD.

 

No matter how much new math you invent, this is not possible. The best thing you can do is approximate it within some bandlimit. That creates "artifacts", such as Gibbs, that are much, much larger than anything a proper windowed FFT would cause. Accept that real world cannot be perfect, and an ideal pulse is not possible. 

You have another option. The 6th one. In time domain it's the piecewise multiplication of time delayed heaviside with same time delayed sine. It is no longer a heaviside. This is continuous function, no jump discontinuities. This is not infinite slew. But this is also transients. You can certainly make this happen in real life. Show me how you'll go about perfectly bandlimiting, sampling and reproducing this without artefacts using sinc filters.

[pkane2001]

Just now, manueljenkin said:

You have another option. The 6th one. In time domain it's the piecewise multiplication of time delayed heaviside with same time delayed sine. It is no longer a heaviside. This is not infinite slew. But this is also transients. You can certainly make this happen in real life. Show me how you'll go about perfectly bandlimiting, sampling and reproducing this without artefacts using sinc filters.

 

No such option. Time domain and frequency domains are equivalent and interchangeable. One cannot exist without the other. That's the Fourier theorem. If you manipulate time domain, you are affecting the frequency domain and vice versa. You don't get a choice to change just one.

[manueljenkin]

Just now, pkane2001 said:

 

No such option. Time domain and frequency domains are equivalent and interchangeable. One cannot exist without the other. That's the Fourier theorem. If you manipulate time domain, you are affecting the frequency domain and vice versa. You don't get a choice to change just one.

You're distorting math to your own opinions. This scenario is perfectly valid.

[pkane2001]

Just now, manueljenkin said:

You're distorting math to your own opinions. This scenario is perfectly valid.

 

Nope. I'm stating exactly what the Fourier theorem states. I'm not re-interpreting it or coming up with new math - you are.

 

[manueljenkin]

12 minutes ago, pkane2001 said:

 

Nope. I'm stating exactly what the Fourier theorem states. I'm not re-interpreting it or coming up with new math - you are.

 

I also said the same thing: Fourier transform decomposition will not work reliably for this signal. But the signal is perfectly valid, just that it cannot be handled properly using Fourier transform. And can't be handled by Nyquist Shannon sampling. Before you twist my words again, this is VALID REAL WORLD SCENARIO, but cannot be handled reliably by Nyquist shannon sampling because a sinc filter cannot perfectly bandlimit this signal.

 

You're adding your own words/context that I never said. And happened multiple times here.

[pkane2001]

25 minutes ago, manueljenkin said:

But the signal is perfectly valid, just that it cannot be handled properly using Fourier transform. And can't be handled by Nyquist Shannon sampling. Before you twist my words again, this is VALID REAL WORLD SCENARIO, but cannot be handled reliably by Nyquist shannon sampling because a sinc filter cannot perfectly bandlimit this signal.

 

So, you are concerned that Shannon-Nyquist cannot be used to perfectly reconstruct an idealized, infinite-slope pulse? Are you just ignoring all my posts that infinities do not exist in the natural world, or do you disagree with this statement? 

 

[manueljenkin]

10 minutes ago, pkane2001 said:

 

So, you are concerned that Shannon-Nyquist cannot be used to perfectly reconstruct an idealized, infinite-slope pulse? Are you just ignoring all my posts that infinities do not exist in the natural world, or do you disagree with this statement? 

 

Did you even read what scenario 6 is. It is NOT INFINITE slope. You said heaviside is infinite slope, and can't be in real existence fine, ignore scenario 5 then. Scenario 6 is just a time delayed sine, that started its journey at time x, the signal being a constant 0 till then. This is a REAL WORLD SCENARIO. Please take some time to read what is mentioned instead of imagining your own intentions. It is evident that you haven't read even 10% of the scenario I presented you, imagining 90% to your thoughts and replying to yourself, twisting what was actually asked in the process.

 

Before @opus101piles up on another disagree button without really knowing what is mentioned, the last 3-4 comments were regards to the scenario 6, not scenario 5. Scenario 5 is the heaviside filter. Scenario 6 is just a time delayed sine starting at time x instead of having begun it's journey at time -infinity.

[manueljenkin]

54 minutes ago, manueljenkin said:

You have another option. The 6th one. In time domain it's the piecewise multiplication of time delayed heaviside with same time delayed sine. It is no longer a heaviside. This is continuous function, no jump discontinuities. This is not infinite slew. But this is also transients. You can certainly make this happen in real life. Show me how you'll go about perfectly bandlimiting, sampling and reproducing this without artefacts using sinc filters.

Here.. tagging the context yet again!! Please read before you speak.

[manueljenkin]

Here is the signal. I've marked it in yellow. There is absolutely no jump discontinuity here.

[pkane2001]

5 minutes ago, manueljenkin said:

Here is the signal. I've marked it in yellow.

 

Sorry, but you'll have to explain this to me in more detail. A delayed sine function cannot be represented by Nyquist-Shannon? Are you serious? That will be big news to every single CD ever recorded that starts with zero level signal and then plays music (sine waves).

[pkane2001]

repeat

[manueljenkin]

7 minutes ago, pkane2001 said:

 

Sorry, but you'll have to explain this to me in more detail. A delayed sine function cannot be represented by Nyquist-Shannon? Are you serious? That will be big news to every single CD ever recorded that starts with zero level signal and then plays music (sine waves).

Do the Fourier transform of this signal (in time domain you represent it as Piecewise multiplication of a time delayed sine and a time delayed heaviside) then multiply it with Fourier transform of sinc. Now tell me what you get, whether it is band limited or not. (This sine frequency is less than fc). Passing this function through a sinc will for sure create out of band components. You cannot bandlimit this using sinc.

 

Only a delay in steady state sine (if it began from time =- infinity and runs through to time +infinity) can be bandlimited properly without having any out of band components during low passing. The above is a transient scenario and it will have out of band components. Especially high right at the point where it stops being dc 0 and starts becoming a sinusoid.